Electrical Resistance 1)I have read that the cause for resistance is charges colliding with atoms. But what exactly are the charges colliding with. Surely charges cannot be straight up hitting the nucleus/electrons.
2) Also if the cause of resistance is purely based on the random( as it seems to me) incident of collision. How is it that we have well defined formulas like V=IR and R=rhol/A.
                      And most importantly Heat energy=VIt. How can we be so sure that all of the loss in PE of charges between 2 points gets converted to heat. How can we say that there will be no gain in KE at all?
Thanks 
 A: 
1) [...] But what exactly are the charges colliding with. Surely charges cannot be straight up hitting the nucleus/electrons.



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*It is easy and intuitive to think of both charges and atoms as spherical balls. The charge-balls collide with the atom-balls.

*If you wish to get closer to reality but still keep an intuitive visualization, then think of a charge (electrons e.g.) as a tiny "spot" in the middle of an electric field "bubble". Likewise with atoms (their electron shells give them a similar negative electric field "bubble"). When a charge gets close to an atom, their fields meet and interact. The charge field might therefore be repelled / pushed. In this way the "collision" is nothing more than a field interaction. The field is like a "pillow" and the collision is like the two pillows bumping together, prevents the charge from ever coming into contact with the atom.

*If you want an even closer-to-reality picture, then the intuitive visualization becomes tough. The electron size is so tiny that quantum probability becomes relevant. Now we can't think of an electron as a spherical ball, but rather as a "probability cloud". This "cloud" carries the charge - this "cloud" is the electron. The cloud is the electron's "soul" and the electron has no "body". The electron is not at a particular point inside the cloud, but there is a certain probability for it to be at each point constituting this cloud. Same goes for each particle of the atom - the electrons in so-called shells should now rather be thought of as several "merged" or "squeezed" "probability clouds", with various shapes depending out the configuration. The attraction/repulsion between this atom and the incoming charge is still the factor that causes deflection of the charge.
All these are more and more detailed models or ways to think. Common for them all is that temperature is defined as the Ångström-scale motion and vibration of atoms.
Anytime a charge interacts with an atom according to any of the above models, that atom is pushed/pulled a bit and gains some kinetic energy. It isn't ripped free from the crystal/molecular structure, so it just vibrates in its place. (Were it ripped free, then that corresponds to the material breaking apart/melting.)
This increased vibrational motion corresponds to a temperature increase, eventually leading to a heat loss to the surroundings. Simultaneously, the charge looses some kinetic energy corresponding to a loss in potential (stored or carries electrical energy).
So, no, there is very rarely any electrons colliding with a nucleus.

2) Also if the cause of resistance is purely based on the random( as it seems to me) incident of collision. How is it that we have well defined formulas like V=IR and R=rhol/A. And most importantly Heat energy=VIt. How can we be so sure that all of the loss in PE of charges between 2 points gets converted to heat. How can we say that there will be no gain in KE at all? Thanks 

Yes, it is random. But random with a "tendency".
We can think of the collisions as being a mix straight-on or sliding off or impacts at different angles, all causing more or less transfer of energy. But there is an average. If you look at 10 electrons, you can find the average collision. If you look at 100 electrons, the average collision is more accurate. If you look at billions and billions of electrons, you are pretty sure that the average is quite accurate. A good representation of the overall collisions that happen.
This is why we can establish relationships such as Ohm's law. The individual electron may not behave strictly according to this law, but the overall average electron will. And therefore the resulting current, consisting of billions and billions of electrons, will.
